Best Li recovery:87.1%

Active experiments:3

Best model:Avrami-Erofeev

Eₐ surface (B,C,D):27.8 kJ/mol

Eₐ diffusion (B,C,D):18.0 kJ/mol

Segmented series:3 / 4

LITIUMLAB

Black mass leaching — kinetic simulation platform

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Experimental data

Dynamic time-series input — add as many time points as your experiment produced

📈

Recovery curves

α vs time with error bars, plateau annotation, curve segmentation

🧮

Model fitting

SCM, Avrami, GB, n-order — AIC/F-test, residuals, parity, segmentation

⚡

Process optimiser

Dual-mechanism aware predictions — sensitivity and economics

Platform overview

Leaching mediumH₂SO₄ (aq)

FeedSpent Li-ion black mass

Kinetic models4

Auto segmentationON

Dual-mech. optimiserINCLUDED

PDF report exportAVAILABLE

Scientific features

Dynamic data entryyes

Recovery curves ± error barsyes

Linearisation + residualsyes

Parity plot + mass balanceyes

Dual-mechanism segmentationyes

Literature Eₐ comparisonyes

Quick links

? Help & User manual→

⚗ Data Input→

⚠ Segmentation detail→

Activation energy→

Process optimiser→

Dashboard

Loading your projects...

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Projects

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Total series

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Segmented

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Fitted

Your projects

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Chemical constants database

System constants + your saved constants — persisted to your account

💾Your constants (green) are saved server-side and reload every time you log in.

All

Thermodynamic

Kinetic

Transport

My constants (2)

Symbol	Name	Value	Units	Category	Source	Action
R	Universal gas constant	8.314	J mol⁻¹ K⁻¹	THERMO	NIST	system
D_Li+	Diffusivity of Li⁺ in H₂SO₄	1.03×10⁻⁹	m² s⁻¹	TRANSPORT	CRC	system
M_Li	Molar mass of lithium	6.941	g mol⁻¹	THERMO	IUPAC	system
ρ_BM	Black mass bulk density	2.85	g cm⁻³	PHYSICAL	User est.	system
E°_LiCoO₂	Standard electrode potential	+0.56	V vs SHE	ELECTROCHEM	Nagaura	system
k_app_BM01	App. rate const. 348K YOURS	0.0414	min⁻¹	KINETIC	Exp. fit
Ea_BM01	Activation energy NMC811 YOURS	42300	J mol⁻¹	KINETIC	Arrhenius

Experimental data input

Define conditions — add as many time points as your experiment produced

Series configuration

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Temperature

K (= 75°C)

H₂SO₄ concentration

mol/L

S/L ratio

g/mL

10g/500mL → 0.020 g/mL

Stirring speed

RPM

Particle size d₅₀

μm

H₂O₂ reductant

vol%

⚖ Mass balance checker

Step 1 — Li in sample:— Step 2 — Acid available:— Step 3 — Acid required:— Result:—

Manual entry

Paste / bulk entry

CSV upload

Enter each time point and recovery fractions. Click + Add time point for each measurement, or press Enter on the last field of a row to jump to a new row.

Rows: 7 | Time: 2–90 min | Max α(Li): 0.870

#	Time (min) required	α Li required · 0–1	α Co optional	α Ni optional	α Mn optional	pH optional	Li s.d. error bars

+

Add time point or press Enter on last field of a row

Copy columns from Excel and paste below. Each row = one time point. Separator: tab, comma, or space. Column order: time_min · α_Li · α_Co · α_Ni · α_Mn · pH · Li_sd. Only first two are required.

Paste data here

Columns: time_min · α_Li · α_Co · α_Ni · α_Mn · pH · Li_sd

⬆

Drop CSV file here or click to browse

Columns: time_min, alpha_Li [, alpha_Co, alpha_Ni, alpha_Mn, pH, alpha_Li_sd]

Expected CSV format

time_min,alpha_Li,alpha_Co,alpha_Ni,alpha_Mn,pH,alpha_Li_sd
2,0.08,0.07,0.06,0.05,1.1,0.005
5,0.21,0.18,0.15,0.12,0.8,0.010

Series metadata

Black mass source / batch

Li content of solid (wt%)

Milling method

Replicate runs (n)

Solid mass used (g)

Solution volume (mL)

Li recovery curves

α vs leaching time — all series

Li extraction fraction (α) vs leaching time — experimentálne dáta + Avrami fit

Plná čiara = Avrami model (najlepší jednofázový fit surových dát) | Body = experimentálne merania ±s.d. | Odchýlky B,C,D od Avrami → bifázový charakter → analýza v module Segmentation

⚑ What is a plateau?A plateau occurs when α stops increasing despite continued leaching. Causes: (1) acid depletion; (2) Li encapsulation in passivation layers; (3) thermodynamic ceiling. At 363K/2.5M: α≈0.91. Use mass balance checker to rule out acid depletion. To raise the plateau: reduce particle size, increase acid, or raise temperature.

Select a project and run fitting to see series results here.

Kinetic model fitting

All 4 models — AIC / F-test / RMSE

Manual breakpoint override: Enable Auto-detection active when disabled

Avrami-Erofeev

[-ln(1-α)]^(1/n) = k·t

★ BEST FIT

R²

0.9620

RMSE

0.0183

AIC

−142.3

n (Avrami)	0.731 ± 0.042
k	0.0414 min⁻¹
Mechanism	Nucleation+growth
Residuals	random ✓

Avrami fit — observed vs predicted (T=348K)

Exp. data ± s.d.

Model fit

95% CI

Shrinking Core

1-(1-α)^(1/3)=k_s·t

R²

0.9410

RMSE

0.0312

k_s	0.039 min⁻¹
k_d	0.009 min⁻¹
F-test p	0.021

Ginstling-Brounshtein

1-2α/3-(1-α)^(2/3)=k_D·t

R²

0.9170

RMSE

0.0490

k_D	0.0072 min⁻¹
Valid α	>0.5
F-test p	0.041

n-order reaction

dα/dt=k·(1-α)^n

R²

0.8930

RMSE

0.0630

n	1.43±0.11
Weak α>0.7	—
F-test p	0.003

Statistical comparison

Model	R²	RMSE	AIC	Rank
Avrami-Erofeev	0.9620	0.0183	−142.3	★ 1st
Shrinking Core	0.9410	0.0312	−118.7	2nd
Ginstling-B.	0.9170	0.0490	−97.1	3rd
n-order	0.8930	0.0630	−82.4	4th

Parity plot — predicted vs observed α

Points on 45° line = perfect prediction.

Linearisation plots

Transform α — straight line confirms model validity

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Activation energy — dual Arrhenius

Eₐ,s and Eₐ,D from segmented series

Click "Calculate Arrhenius" to compute activation energies from all biphasic series in the current project.

Requires at least 3 series with detected breakpoints.

Process optimiser

Dual-mechanism aware — sliders produce real predictions in both modes

Reference: Arrhenius analysis Single series:

Parameter controls

Use dual-mechanism model ON

Sensitivity analysis

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Predicted Li recovery (α)

Predicted α vs time

Stage 1 Stage 2 | Breakpoint

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Acid consumed (mol/mL)

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Est. acid cost (€/kg Li)

Mechanism detail

Select reference and run prediction

PDF report export

Publication-ready summary — structured like journal supplementary data

Dual-mechanism segmentation

Segmentation results

Loading segmentation data...

Help & User manual

Complete guide — from first data entry to publication-ready results

01

Step-by-step workflow

1

Register and log in

Create an account — your projects, series and all results are stored permanently server-side. Admin must approve your account before first login.

2

Create a project and series in Dashboard

Go to Dashboard → New project. Then create a series for each experimental condition (temperature, acid, S/L, RPM, particle size). Each series will hold one set of α vs t measurements.

3

Input experimental data

Select a series in Dashboard — it opens Data Input automatically with series conditions pre-filled. Enter α(Li) vs time data. Three methods: manual row-by-row, paste from Excel (bulk entry), or CSV upload. Include t=0, α=0 as first point for best fitting results.

4

Check mass balance

Mass balance checker calculates automatically from your conditions. It shows: (1) moles of Li in sample, (2) moles of acid available, (3) moles required. If excess is below 2×, the plateau on your curve may be caused by acid depletion — not by kinetics.

5

Save and visualise

Click "Save & visualise →" to store data and go to Recovery Curves. Check temperature trend and curve shape across all series.

6

Run model fitting

Go to Model Fitting. The system automatically fits all kinetic models and selects the best using AIC (Akaike Information Criterion). If a biphasic mechanism is detected, it fits SCM (surface) before the breakpoint and GB (diffusion) after. You can override the breakpoint manually if needed.

7

Check linearisation

Go to Linearisation. A straight line confirms the model is valid. A curved line means the model is wrong even if R² looks acceptable. Use this to visually validate model selection.

8

Calculate activation energy

Go to Activation Energy. Run at least 3 series at different temperatures. The system computes Eₐ from linearisation-derived k values using the Arrhenius equation. For biphasic series: dual Arrhenius gives Eₐ,s (surface) and Eₐ,D (diffusion). You can select which series to include.

9

Use the Optimiser

Go to Optimiser. Adjust temperature, particle size, acid concentration, S/L ratio, stirring speed and leach time using sliders. The system predicts α using the Arrhenius-corrected rate constants from your best series. Dual mechanism ON uses SCM+GB model; OFF uses single Avrami.

10

Export report

Go to Report Export. Select sections, choose journal style (Generic, Elsevier, ACS, RSC), and generate a PDF with all results including graphs, tables and raw data appendix.

02

Data input — three methods

Manual entry: Type directly into the table. Click "+ Add time point" or press Enter on the last field to create a new row. Sort by time ↑ reorders rows.

Paste / bulk entry: Copy columns from Excel and paste into the text area. Accepts tab, comma, or space separated values. Column order: time_min · α_Li · α_Co · α_Ni · α_Mn · pH · Li_sd. Click "Import →" then review in Manual tab.

CSV upload: Upload a .csv file. Required columns: time_min, alpha_Li. Optional: alpha_Co, alpha_Ni, alpha_Mn, pH, alpha_Li_sd. Data is sorted by time automatically after import.

03

Model fitting — how it works

The fitting algorithm uses AIC model selection — it fits all models simultaneously and picks the best one objectively.

Step 1 — Single-phase models fitted:
• Avrami-Erofeev: α = 1 − exp(−k·tⁿ) — nucleation and growth
• SCM (full curve): 1−(1−α)^(1/3) = k_s·t — surface reaction
• GB (full curve): 1−(2/3)α−(1−α)^(2/3) = k_D·t — diffusion control

Step 2 — Biphasic models tested at every possible breakpoint:
• SCM→GB: SCM before t_bp, GB after t_bp (most common)
• SCM→SCM: two different surface reaction rates

Step 3 — AIC selects winner:
AIC = n·ln(RSS/n) + 2·p, where p = number of parameters. Biphasic model wins only if its AIC is at least 2 units lower than the best single-phase model — this prevents false detection.

Manual breakpoint override: If the algorithm misses a visually obvious breakpoint (e.g. too few early samples), you can set t_bp manually in the yellow banner on Model Fitting page. Each series stores its own manual t_bp — it persists between sessions and is used by all modules automatically.

04

Activation energy — Arrhenius analysis

k values are derived from linearisation — not from curve_fit — for consistency and transparency.

For biphasic series (≥3 required):
• k_s from SCM linearisation of points before t_bp
• k_D from GB linearisation of points after t_bp
• Dual Arrhenius: separate Eₐ,s and Eₐ,D

For single-phase series (≥3 required, same model):
• k from the best linearisation model (Avrami, SCM, or GB)
• Single Arrhenius: one Eₐ

Mix of types → warning shown. You can select which series to include using checkboxes in the results table. Minimum 2 series allowed (with warning); minimum 3 recommended.

Physical check: Eₐ,s should be greater than Eₐ,D — surface reaction is always more temperature sensitive than diffusion. If not, check your t_bp settings.

Typical values for H₂SO₄ leaching of black mass: Eₐ,s = 20–60 kJ/mol, Eₐ,D = 5–25 kJ/mol.

05

Scientific background — kinetic models

Shrinking Core Model (SCM) — surface reaction

1 − (1 − α)^(1/3) = k_s · t

Particle shrinks as a sphere. Rate limited by surface chemical reaction. Linear in t — slope = k_s. Valid in early stage of leaching before product layer builds up.

Ginstling-Brounshtein (GB) — diffusion

1 − (2/3)α − (1 − α)^(2/3) = k_D · t

Diffusion through a growing product layer. Rate decreases as the layer thickens. Linear in t — slope = k_D. Valid after the surface reaction is complete (second stage of biphasic mechanism).

Avrami-Erofeev

α = 1 − exp(−k·tⁿ) → ln[−ln(1−α)] = n·ln(k) + n·ln(t)

Nucleation and growth model. n ≈ 0.5–1: diffusion-controlled growth. n ≈ 1–2: interface-controlled. n ≈ 2–3: 2D/3D growth. Linearised form uses ln(t) on x-axis because t appears inside the power n. Slope = n, intercept = n·ln(k).

Arrhenius equation

k(T) = A · exp(−Eₐ / R·T) → ln(k) = ln(A) − Eₐ/(R·T)

A = pre-exponential factor, Eₐ = activation energy (kJ/mol), R = 8.314 J·mol⁻¹·K⁻¹. Plot ln(k) vs 1000/T — slope = −Eₐ/R. Higher Eₐ means stronger temperature dependence.

Biphasic (dual) mechanism

Many real leaching systems show two distinct kinetic regimes: fast surface reaction followed by slower diffusion through the product layer. The breakpoint t_bp is where the controlling mechanism switches. Before t_bp: SCM controls (k_s). After t_bp: GB diffusion controls (k_D). This gives two separate Eₐ values from Arrhenius analysis.

06

Module reference

Dashboard →

Create and manage projects and series. Click a project to expand its series list. Click a series to open it in Data Input. Delete series or projects with the Del button.

Data Input →

Enter α vs t data. Series conditions load automatically from Dashboard. Mass balance checker updates live. Three entry methods: manual, bulk paste, CSV upload.

Recovery Curves →

Visualise all series α vs t. Toggle between all series and current series. Check for biphasic shape (fast rise then plateau-like slowdown).

Model Fitting →

Automatic AIC model selection. Shows fitted curve, R², AIC, k values. Manual breakpoint override available in yellow banner — each series remembers its own t_bp. Use "All series" button to compare all series side by side.

Linearisation →

Three linearisation plots: Avrami (ln[-ln(1-α)] vs ln(t)), SCM (1-(1-α)^1/3 vs t), GB (f_GB(α) vs t). Straight line = model valid. R² table shows best model for each series.

Activation Energy →

Arrhenius analysis from linearisation-derived k values. Automatic method selection (dual for biphasic, single for single-phase). Series selection checkboxes. Minimum 2 series (3 recommended).

Segmentation →

Detailed biphasic analysis. Shows breakpoint, per-segment fits and rate constants. Confidence level (high/medium) based on AIC delta.

Optimiser →

Predicts α for any combination of process conditions. Uses Arrhenius-corrected k values from best series as baseline. Dual ON: SCM+GB model. Dual OFF: Avrami. Sensitivity bars show which parameter has the most impact.

Report Export →

Generate PDF report. Select sections (conditions, recovery curves, model fitting, linearisation, segmentation, Arrhenius, optimiser, raw data). Choose journal style and generate.

07

Frequently asked questions

How many temperature points do I need?

Minimum 3, recommended 4–5. Spread evenly — e.g. 298K, 323K, 348K, 363K. More temperatures give more reliable Eₐ with smaller confidence intervals.

How many time points per experiment?

Minimum 5, recommended 7–10. Cover early kinetics through to plateau. For detecting biphasic mechanism, sample densely in the first few minutes — this is where the surface reaction occurs and where the breakpoint is.

The algorithm says single-phase but my curve looks biphasic — what do I do?

Use the manual breakpoint override in Model Fitting (yellow banner). Enable it, enter the time where you see the slope change, and click Apply. The t_bp is saved for that series and used by all modules including Arrhenius. This is common when you have few early-time samples and the AIC test cannot distinguish the two phases statistically.

Why does Arrhenius give different Eₐ depending on which series I include?

Because k values must be comparable between series — same model, same mechanism. Use the checkboxes to exclude outlier series. Series with R² below 0.85 or non-monotonic k values should be excluded or their t_bp re-checked.

What is S/L ratio and what units?

S/L = solid mass (g) / solution volume (mL). Example: 10 g / 500 mL = 0.020 g/mL. Valid range: 0.001–0.5 g/mL for typical leaching.

My curve does not reach α=1 — is this a problem?

No — this is normal. The plateau represents the practical maximum under those conditions. Check mass balance first. If acid is in excess but plateau is below 1, it is caused by unreactive Li phases, particle encapsulation, or equilibrium limitation.

Should I include t=0, α=0 in my data?

Yes — always include t=0, α=0 as the first data point. It anchors the curve at the origin and improves fitting quality, especially for Avrami and SCM models.

What does AIC mean and why use it?

AIC (Akaike Information Criterion) balances goodness of fit against model complexity. A model with more parameters (like biphasic) must fit significantly better to win — otherwise the simpler model is preferred. This prevents false detection of biphasic mechanism when data is noisy or sparse.

08

Tips for better results

Sample early and often. The first 5–10 minutes are critical for detecting the surface reaction phase. Take samples at t = 0.5, 1, 2, 5, 10 min before extending to longer times. Without early data, the algorithm cannot reliably detect biphasic mechanism.

Check k values monotonicity. Before running Arrhenius, verify that k_s and k_D increase with temperature. Non-monotonic k values indicate inconsistent t_bp settings across series — adjust manual breakpoints.

Use consistent conditions. For Arrhenius, vary only temperature between series — keep acid, S/L, RPM and particle size constant. Varying multiple parameters makes Eₐ physically meaningless.

Validate with linearisation. Always check the linearisation plots after model fitting. A curved line in the linearisation plot means the model is inappropriate — even if R² is high. R² can be misleading; the linearisation plot is the true test.

Do not over-acidify. Once α plateaus, more acid does not improve recovery but increases neutralisation cost downstream. 2–4× stoichiometric excess is usually optimal.

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